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Determine associativity and commutativity
Given:
\( a * b = \frac{a + b}{2}, \quad a,b \in \mathbb{Q} \)
Commutativity:
\( a * b = \frac{a+b}{2} = \frac{b+a}{2} = b * a \)
✔ Operation is commutative
Associativity:
Check LHS:
\( (a*b)*c = \left(\frac{a+b}{2} * c\right) = \frac{\frac{a+b}{2} + c}{2} = \frac{a+b+2c}{4} \)
Check RHS:
\( a*(b*c) = \left(a * \frac{b+c}{2}\right) = \frac{a + \frac{b+c}{2}}{2} = \frac{2a+b+c}{4} \)
Since:
\( \frac{a+b+2c}{4} \neq \frac{2a+b+c}{4} \)
❌ Operation is NOT associative
Conclusion:
✔ Commutative but ❌ Not associative